To me it sounds like you are saying no, an employer cannot compel you to give them your personal information, but that they can terminate your employment if you refuse to hand it over
is that what you meant? I think 08-15's concern was more along the lines of an employer having the ability to demand someone to give up their personal information without their consent.
Considering this is an American federal court case between two American companies, concerning an American employee, for actions in America, I don't think "Yes but in Europe..." is relevant.